European Congress on Computational Methods in Applied ... - MGNet

Craig C. Douglas and Karen M. Walterscorner points to approximate the central point. They are calculated as follows:u 2i+1,2j+1 = 1 4 (u 2i,2j + u 2i+2,2j + u 2i,2j+2 + u 2i+2,2j+2 +2h 2 f 2i+1,2j+1 )+O(h 4 ).The points (x 2i ,y 2j+1 )and(x 2i+1 ,y 2j ) use the standard five-point difference scheme, calledthe + scheme. The calculation of these points involves the points at which the solutionis known and the × points:u k,l = 1 4 (u k−1,l + u k+1,l + u k,l−1 + u k,l+1 + h 2 f k,l )+O(h 4 ),where k =2i +1,l =2j or k =2i, l =2j +1.The projection used in the V cycle is the full weighted restriction:u (j−1)i,j = 116 [u(j) 2i−1,2j−1 + u (j)2i−1,2j+1 + u (j)2i+1,2j−1 + u (j)2i+1,2j+1+2(u (j)2i,2j−1 + u (j)2i,2j+1 + u (j)2i−1,2j + u (j)2i+1,2j)+4u (j)2i,2j].The second order discretization involves the standard five-point formula:4u i,j − u i−1,j − u i+1,j − u i,j−1 − u i,j+1 = h 2 f i,jThe fourth order discretization uses the nine-point formula known as Mehrstellenverfahren[4]:20u i,j − 4(u i−1,j + u i+1,j + u i,j−1 + u i,j+1 ) − (u i−1,j−1 + u i−1,j+1 + u i+1,j−1 + u i+1,j+1 )= h22 (8f i,j + f i−1,j + f i+1,j + f i,j−1 + f i,j+1 )We give a comparison of the lower and higher order methods for Poisson’s problem intwo dimensions for a model problem that has the solution e xy sin(πx)sin(πy). The numberof iterations on each level is given for the O(h 2 ) discretization with linear interpolationand the O(h 4 ) discretization with fourth order interpolation. The coarsest level is level 1where the number of grid points in each direction is equal to seven.Method Level2 3 4 5Lower order 10 6 4 3Higher order 4 2 1 1Table 1: Comparisons for the two dimensional model problem8